25 hard SAT Area and volume questions

Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.

Math · Geometry and Trigonometry~2 per testHard tier

New to this skill? Read the Area and volume strategy guide first: what it tests, how it is scored, and the traps below in plain English.

What makes these hard

Scales area linearly (×k) instead of ×k².
Mixes radius and diameter.
Forgets π or a fractional constant in a volume formula.

Question 1Hard

In the xy-plane, a trapezoid has vertices at (0, 0), (10, 0), (8, 5), and (2, 5). What is the area, in square units, of this trapezoid?

Show the answer and explanation

Why C is right

The trapezoid has parallel bases of length 10 (from (0, 0) to (10, 0)) and length 6 (from (2, 5) to (8, 5)), with height 5. Using the trapezoid area formula $\displaystyle A = \frac{1}{2} (b_{1} + b_{2})$ h: $\displaystyle A = \frac{1}{2}$ (10 + 6)(5) = (1/2)(16)(5) = 40.

Why the others are wrong

AThis omits the factor of 1/2 from one of the dimensions or miscalculates the base lengths.
BThis approximates the perimeter or incorrectly adds dimensions without using the proper area formula.
DThis computes the area of a rectangle with dimensions 10 by 5, not accounting for the trapezoidal shape.

Question 2Hard

A rectangle has a length of 12 centimeters and a width of 5 centimeters. A second rectangle is similar to the first rectangle, and the width of the second rectangle is 20 centimeters. What is the area, in square centimeters, of the second rectangle?

Show the answer and explanation

Why C is right

The width scales from 5 to 20, so the scaling factor is $k = 4$ . The area of the first rectangle is 12 × 5 = 60 square centimeters. Since area scales by $k^{2}$ , the area of the second rectangle is $60 \times 4^{2} = 60 \times 16 = 960$ square centimeters.

Why the others are wrong

AThis incorrectly multiplies the first rectangle's area by 4 instead of by 4².
BThis incorrectly multiplies the first rectangle's area by 8 (which is 2 × 4) instead of by 4².
DThis incorrectly applies the scaling factor to the perimeter (34 × 4 = 136, close to this value) instead of to area.

Question 3Hard

A cube has an edge length of 5 inches. A second cube is similar to the first cube, and the volume of the second cube is 1000 cubic inches. What is the surface area, in square inches, of the second cube?

Show the answer and explanation

Why A is right

The volume of the first cube is $5^{3} = 125$ cubic inches. The second cube has volume 1000 cubic inches, so the volume scales by $1000/125 = 8 = 2^{3}$ , meaning $k = 2$ . The surface area of the first cube is $6(5^{2})$ = 150 square inches. Surface area scales by $k^{2}$ , so the surface area of the second cube is $150 \times 2^{2} = 150 \times 4 = 600$ square inches.

Why the others are wrong

BThis incorrectly uses only the area of four faces or applies the wrong scaling factor.
CThis incorrectly multiplies the first cube's surface area by 2 instead of by 2².
DThis incorrectly applies calculations related to perimeter or edge sums rather than surface area.

Question 4Hard

A circle has radius r and area $49\pi$ . A square is inscribed in this circle such that all four vertices of the square lie on the circle. What is the area of the square?

Show the answer and explanation

Why B is right

From $\pi r^{2}$ $= 49\pi$ , we get $r^{2}$ $= 49$ , so $r = 7$ . For a square inscribed in a circle, the diagonal of the square equals the diameter of the circle, which is $2r = 14$ . If the square has side length s, then $\displaystyle s\sqrt{2}$ $= 14$ , so $s^{2}$ $= 196/2 = 98$ .

Why the others are wrong

AThis incorrectly assumes the area of the square equals r² = 49, not accounting for the inscribed relationship.
CThis incorrectly uses the diameter squared, (2r)² = 196, as the area of the square instead of recognizing the diagonal-side relationship.
DThis results from incorrectly computing 2 × 196 = 392, possibly doubling the incorrect answer.

Question 5Hard

A square has side length s. The square is inscribed in a circle such that all four vertices of the square lie on the circle. The area of the circle is how many times the area of the square?

Show the answer and explanation

Why B is right

The square has area $s^{2}$ . The diagonal of the square is $\displaystyle s\sqrt{2}$ , which equals the diameter of the circle, so the radius is $\displaystyle s\sqrt{2} /2$ . The circle's area is $\pi(s\surd[MATH]2/2)$ ^{2} = \pi $(s^{2}[MATH]\cdot[MATH]2/4)$ [/MATH][/MATH] = \pi s^{2} /2[/MATH]. The ratio is $\pi s^{2} /2$ divided by $s^{2}$ , which equals $\pi /2$ .

Why the others are wrong

AThis inverts the relationship or incorrectly relates the square's side to the circle's radius.
CThis omits the factor of 1/2 that arises from the diagonal-to-radius relationship.
DThis doubles the correct ratio, possibly from confusing diameter with radius in the calculation.

Question 6Hard

A right triangle has legs of length 9 centimeters and 12 centimeters. A second right triangle is similar to the first, and its hypotenuse is 25 centimeters. What is the area, in square centimeters, of the second triangle?

Show the answer and explanation

Why C is right

The first triangle has legs 9 and 12, so its hypotenuse is √ $([MATH]9^{2} + 12^{2})$ [/MATH] = $\displaystyle \sqrt{81 + 144} = \sqrt{225} = 15$ centimeters. The linear scale factor from the first to the second triangle is $25/15 = 5/3$ . The area of the first triangle is $\displaystyle \frac{1}{2}(9)(12) = 54$ square centimeters. The area of the second triangle is $\displaystyle 54 \times \frac{5}{3}^{2} = 54 \times 25/9 = 150$ square centimeters.

Why the others are wrong

AThis is the area of the first triangle, not the second triangle.
BThis uses the linear scale factor 5/3 directly instead of squaring it to get the area scale factor.
DThis incorrectly doubles the correct area or uses an incorrect calculation method.

Question 7Hard

A rectangular garden has a length of 18 feet and a width of 12 feet. A second rectangular garden has dimensions such that each dimension is $\displaystyle \frac{2}{3}$ the corresponding dimension of the first garden. What is the area, in square feet, of the second garden?

Show the answer and explanation

Why C is right

The first garden has area 18 × 12 = 216 square feet. When linear dimensions are scaled by $k = 2/3$ , area scales by $k^{2}$ = $\displaystyle \frac{2}{3}^{2} = 4/9$ . The second garden has area 216 × 4/9 = 96 square feet.

Why the others are wrong

AThis incorrectly applies the scale factor three times instead of twice, computing 216 × (2/3)³.
BThis results from incorrectly computing 216 × 2/3 = 144, then dividing by 2.
DThis incorrectly uses the linear scale factor (216 × 2/3) instead of the area scale factor.

Question 8Hard

A cube has edge length 6 inches. A second cube has a volume that is 125 times the volume of the first cube. What is the edge length, in inches, of the second cube?

Show the answer and explanation

Why B is right

The first cube has volume $6^{3} = 216$ cubic inches. The second cube has volume 125(216) = 27000 cubic inches. The edge length is the cube root of 27000, which is 30 inches. Alternatively, since volume scales by $k^{3}$ , we have $k^{3}$ = 125, so $k = 5$ , and the edge length is 6(5) = 30.

Why the others are wrong

AThis results from incorrectly taking the square root of 125 and adding it to the original edge length.
CThis represents 6 times the original edge, applying the linear multiplier incorrectly.
DThis results from multiplying 125 by 6 without applying cube root reasoning.

Question 9Hard

A figure consists of a square with side length 12 and a quarter-circle of radius 12 in one corner of the square removed. What is the area of the figure?

Show the answer and explanation

Why C is right

The area of the square is $12^{2} = 144$ . The quarter-circle has radius 12, so its area is $\displaystyle \frac{1}{4}\pi (12)^{2} = 36\pi$ . Since the quarter-circle is removed from the square, the area of the figure is $144 - 36\pi$ .

Why the others are wrong

AThis adds the quarter-circle area instead of subtracting it, treating the removal as an addition.
BThis uses the full circle formula π(12)² = 144π instead of the quarter-circle formula (1/4)π(12)².
DThis computes the quarter-circle area as (1/2) × 144 = 72 instead of (1/4) × 144π, confusing the fractional factor.

Question 10Hard

A cube has edge length 4. A second cube has a volume that is 8 times the volume of the first cube. What is the edge length of the second cube?

Show the answer and explanation

Why B is right

The volume of the first cube is $4^{3} = 64$ . The second cube has volume $8 \times 64 = 512$ . Since volume = $\text{edge}^{3}$ , the edge of the second cube is $\displaystyle \sqrt[3]{512} = 8$ . Alternatively, if volume scales by 8, the linear scale factor is $\displaystyle \sqrt[3]{8} = 2$ , so edge length is $4 \times 2 = 8$ .

Why the others are wrong

AThis incorrectly adds the volume ratio to the original edge rather than applying the cube root.
CThis incorrectly doubles the correct answer, possibly from confusing the volume scaling with linear scaling.
DThis applies the volume ratio directly to the edge length without taking the cube root.

Question 11Hard

Triangle PQR is formed by connecting the points P(2, 6), Q(10, 6), and R(10, 0) in the xy-plane. What is the area, in square units, of triangle PQR?

Show the answer and explanation

Why B is right

The triangle has vertices P(2, 6), Q(10, 6), and R(10, 0). The base PQ is horizontal with length 10 - 2 = 8, and the height from R perpendicular to PQ is 6 - 0 = 6. Using $\displaystyle A = \frac{1}{2}$ bh gives $\displaystyle A = \frac{1}{2}$ (8)(6) = 24.

Why the others are wrong

AThis results from forgetting the factor of 1/2 in the triangle area formula and then dividing by 2 incorrectly, or miscalculating dimensions.
CThis results from calculating the perimeter (8 + 6 + 10 = 24) and adding an extra 4, confusing perimeter with area.
DThis results from multiplying base and height without the factor of 1/2 (8 × 6 = 48), using the rectangle area formula instead.

Question 12Hard

A sphere has a radius of r centimeters. A second sphere has a radius of 4r centimeters. The surface area of the second sphere is how many times the surface area of the first sphere?

Show the answer and explanation

Why C is right

The surface area of a sphere is $4\pi r^{2}$ . For the first sphere, S $A_{1}$ $= 4\pi r^{2}$ . For the second sphere with radius $4r$ , S $A_{2}$ $= 4\pi (4r)^{2} = 4\pi (16r^{2})$ $= 64\pi r^{2}$ . The ratio is $64\pi r^{2} /(4\pi[MATH]r^{2}[MATH])$ [/MATH] = 16[/MATH]. Alternatively, since surface area scales by the square of the linear scale factor: $4^{2} = 16$ .

Why the others are wrong

AThis is the linear scale factor, but surface area scales by the square of the linear scale factor.
BThis incorrectly applies 2 times the linear scale factor, which doesn't correspond to the proper geometric scaling.
DThis is the volume scale factor (4³ = 64), but surface area scales by the square of the linear scale factor, not the cube.

Question 13Hard

In the xy-plane, a triangle has vertices at (0, 0), (8, 0), and (8, 15). A second triangle has vertices at (0, 0), (4, 0), and (4, k), where k is a positive constant. The area of the second triangle is one fourth the area of the first triangle. What is the value of k?

Show the answer and explanation

Why B is right

The first triangle has base 8 and height 15, so its area is (1/2)(8)(15) = 60. The second triangle must have area 60/4 = 15. With base 4 and height k, (1/2)(4) $(k) = 15$ , so $2k = 15$ and $k = 7.5$ .

Why the others are wrong

AThis results from incorrectly dividing 15 by 4 instead of properly setting up the area equation for the second triangle.
CThis incorrectly assumes that if the base is scaled by 1/2, the height should remain the same, not recognizing that the area constraint requires the height to also change.
DThis results from incorrectly setting up the perimeter relationship instead of the area relationship, leading to doubling instead of the correct calculation.

Question 14Hard

A figure is formed by a trapezoid with parallel sides of lengths 20 and 14 and height 8, with a circular region of radius 3 removed from its interior. What is the area of the figure?

Show the answer and explanation

Why B is right

The area of the trapezoid is $\displaystyle \frac{1}{2}(20 + 14)(8) = \frac{1}{2}(34)(8) = 136$ . The circular region has radius 3, so area $\pi (3)^{2} = 9\pi$ . Since the circle is removed, the area of the figure is $136 - 9\pi$ .

Why the others are wrong

AThis adds the circular area instead of subtracting it, treating the removal as an addition to the trapezoid.
CThis uses the diameter 6 as the radius in the circle formula: π(6)² = 36π.
DThis doubles the correct circle area 9π to get 18π, possibly confusing the radius with a diameter-based calculation.

Question 15Hard

A right triangle has legs of length 9 and 12. A second right triangle is similar to the first triangle, and the hypotenuse of the second triangle is 5 times the hypotenuse of the first triangle. The area of the second triangle is how many times the area of the first triangle?

Show the answer and explanation

Why C is right

The hypotenuse of the first triangle is √ $([MATH]9^{2} + 12^{2})$ [/MATH] = $\displaystyle \sqrt{81 + 144} = 15$ . For similar triangles, if the hypotenuse is scaled by factor k, all linear dimensions scale by k, and area scales by $k^{2}$ . Since $k = 5$ , the area of the second triangle is $5^{2} = 25$ times the area of the first.

Why the others are wrong

AThis uses the linear scale factor instead of its square for area scaling.
BThis incorrectly applies an intermediate calculation or confuses area with a mixed dimensional measure.
DThis incorrectly applies the volume scaling factor k³ instead of the area scaling factor k².

Question 16Hard

In the xy-plane, a rectangle has vertices at (0, 0), (6, 0), (6, 4), and (0, 4). What is the area, in square units, of a rectangle with vertices at (0, 0), (3, 0), (3, 2), and (0, 2)?

Show the answer and explanation

Why A is right

The second rectangle has length 3 and width 2, giving area (3)(2) = 6. Each dimension of the second rectangle is half the corresponding dimension of the first rectangle, so area scales by $\displaystyle \frac{1}{2}^{2} = 1/4$ . The first rectangle has area (6)(4) = 24, so the second has area 24/4 = 6.

Why the others are wrong

BThis incorrectly computes the perimeter of the second rectangle, 2(3 + 2) = 10, instead of its area.
CThis incorrectly applies linear scaling k = 1/2 instead of area scaling k² = 1/4, computing 24/2 = 12.
DThis gives the area of the first rectangle instead of the second rectangle.

Question 17Hard

A right triangle has legs of length 9 centimeters and 12 centimeters. A second right triangle is similar to the first triangle, and the hypotenuse of the second triangle has length 5 centimeters. What is the area, in square centimeters, of the second triangle?

Show the answer and explanation

Why B is right

The first triangle has legs 9 and 12, so its hypotenuse is $\displaystyle \sqrt{81 + 144} = \sqrt{225} = 15$ centimeters. The scale factor from the first to the second triangle is 5/15 = 1/3. The area of the first triangle is (1/2)(9)(12) = 54 square centimeters. Since area scales by the square of the linear scale factor, the area of the second triangle is $\displaystyle 54 \times \frac{1}{3}^{2} = 54 \times \frac{1}{9} = 6$ square centimeters.

Why the others are wrong

AThis results from using an incorrect scale factor relationship or halving the correct area value inappropriately.
CThis results from confusing the scale factor or incorrectly applying the area scaling relationship.
DThis results from using a perimeter-based approach or incorrectly relating the dimensions without squaring the scale factor.

Question 18Hard

Circle P has a radius of 5. Circle Q has an area that is 16 times the area of circle P. What is the radius of circle Q?

Show the answer and explanation

Why C is right

Circle P has area $\pi (5^{2})$ $= 25\pi$ . Circle Q has area $16([MATH]25\pi )$ = 400\pi[/MATH]. Since area $= \pi r^{2}$ , we have $\pi r^{2}$ $= 400\pi$ , so $r^{2}$ $= 400$ and $r = 20$ .

Why the others are wrong

AThis incorrectly takes the linear scaling factor as the square root of half of 16, or simply doubles the original radius.
BThis uses 16 directly as the radius, confusing the area scaling factor with the radius itself.
DThis multiplies 5 by 16 to get 80, treating the area scaling factor as a linear scaling factor.

Question 19Hard

A rectangular garden has a length of 18 meters and a width of 12 meters. A concrete border that is 2 meters wide surrounds the garden on all sides. What is the area, in square meters, of the concrete border?

Show the answer and explanation

Why A is right

The outer dimensions are (18 + 2×2) by (12 + 2×2) = 22 by 16 meters. The outer area is 22 × 16 = 352 square meters. The inner garden area is 18 × 12 = 216 square meters. The border area is 352 − 216 = 136 square meters.

Why the others are wrong

BThis results from calculating the perimeter of the border instead of its area.
CThis results from doubling the correct area.
DThis results from dividing the correct area by 2.

Question 20Hard

A cylinder has a radius of 7 centimeters and a height of 15 centimeters. A second cylinder has a radius of 14 centimeters and a height of 30 centimeters. The lateral surface area of the second cylinder is how many times the lateral surface area of the first cylinder?

Show the answer and explanation

Why B is right

The lateral surface area of a cylinder is $2\pi rh$ . Each linear dimension (radius and height) of the second cylinder is 2 times the corresponding dimension of the first. When all linear dimensions scale by k, area scales by $k^{2}$ . Therefore, the lateral surface area scales by $2^{2} = 4$ .

Why the others are wrong

AThis is the linear scale factor, not the area scale factor. Surface area scales by the square of the linear scale factor.
CThis incorrectly applies k³ (volume scaling) instead of k² (area scaling).
DThis applies k⁴ instead of k² for the area scaling relationship.

Question 21Hard

In the xy-plane, a triangle has vertices at (-4, 1), (8, 1), and (2, 9). What is the area, in square units, of this triangle?

Show the answer and explanation

Why C is right

The base of the triangle lies on the horizontal line $y = 1$ , from $x = -4$ to $x = 8$ , giving base length 12. The height is the vertical distance from $y = 1$ to $y = 9$ , which is 8. The area is (1/2)(12)(8) = 48 square units.

Why the others are wrong

AThis results from incorrectly computing dimensions or applying an incorrect formula.
BThis results from using incorrect base or height values in the area calculation.
DThis incorrectly omits the factor of 1/2 in the triangle area formula, computing 12 × 8 = 96.

Question 22Hard

A right circular cone has a base radius of 6 centimeters and a height of 8 centimeters. A second right circular cone has a base radius of 18 centimeters and a height of 24 centimeters. The volume of the second cone is how many times the volume of the first cone?

Show the answer and explanation

Why D is right

The volume of a cone is $\displaystyle V = \frac{1}{3} \pi r^{2}$ h. For the first cone, $V_{1}$ $\displaystyle = \frac{1}{3}\pi(6) ^{2} (8) = 96\pi$ . For the second cone, $V_{2}$ $\displaystyle = \frac{1}{3}\pi(18) ^{2} (24) = \frac{1}{3}\pi(324) (24) = 2592\pi$ . The ratio is $2592\pi /96\pi = 27$ . Each linear dimension is scaled by $k = 3$ , so volume scales by $k^{3}$ $= 27$ .

Why the others are wrong

AThis represents the linear scaling factor only, not accounting for the cubic relationship between linear dimensions and volume.
BThis represents k² where k = 3, which would apply to area scaling but not to volume scaling.
CThis incorrectly calculates 2k² = 2(9) = 18, mixing linear and quadratic scaling factors.

Question 23Hard

Square A has a side length of 6 meters. Square B has a side length that is k times the side length of square A, where k is a positive constant. The area of square B is 324 square meters. What is the value of k?

Show the answer and explanation

Why A is right

Square A has area $6^{2} = 36$ square meters. Square B has side length 6k, so its area is $(6k)^{2} = 36k^{2}$ . Setting this equal to 324 gives $36k^{2} = 324$ , so $k^{2}$ = 9, and $k = 3$ .

Why the others are wrong

BThis incorrectly assumes the area ratio equals the side length ratio, computing 324/36 = 9 and taking the linear factor instead of the square root.
CThis is the ratio of the areas (324/36 = 9) rather than the ratio of the side lengths, which is the square root of the area ratio.
DThis divides the side length of square B by 2 instead of using the proper scaling relationship.

Question 24Hard

A cylinder has radius r and height h. A second cylinder has radius 2r and height h. The lateral surface area of the second cylinder is how many times the lateral surface area of the first cylinder?

Show the answer and explanation

Why A is right

The lateral surface area of a cylinder is $2\pi rh$ . For the first cylinder this is $2\pi rh$ . For the second cylinder with radius $2r$ and height h, the lateral surface area is $2\pi(2r) h = 4 \pi rh$ , which is 2 times the first cylinder's lateral surface area.

Why the others are wrong

BThis incorrectly applies the area scaling factor k² to a situation where only one dimension changed.
CThis incorrectly includes π in the ratio, which cancels when comparing the two surface areas.
DThis incorrectly applies volume scaling k³ instead of considering how lateral surface area changes.

Question 25Hard

A right circular cylinder has a height of 12 and a radius of r. A second right circular cylinder has a height of 12 and a radius of 3r. The volume of the second cylinder is how many times the volume of the first cylinder?

Show the answer and explanation

Why C is right

The volume of a cylinder is $\pi r^{2}$ h. The first cylinder has volume $\pi([MATH]r^{2}[MATH])$ [/MATH] (12) = 12\pi r^{2}[/MATH]. The second cylinder has volume $\pi(3r) ^{2} (12) = \pi (9r^{2}) (12) = 108\pi r^{2}$ . Dividing $108\pi r^{2}$ by $12\pi r^{2}$ gives 9.

Why the others are wrong

AThis incorrectly uses only the linear scaling factor k = 3, not recognizing that volume scales by k³ for 3D figures when only one dimension (radius) is scaled while height remains constant, leading to k² scaling for volume of cylinders.
BThis results from incorrectly doubling the linear factor, possibly from computing 3 × 2 instead of applying the correct area scaling relationship.
DThis incorrectly applies k³ = 27 as if all three dimensions were scaled, but only the radius is scaled by 3 while height remains the same.

These 25 are a sample, not a study plan

These 25 come from a bank of 17,599 questions across all 31 SAT skills and three difficulty tiers. Inside SAT Climb you get the rest of Area and volume, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.

Start free →Area and volume strategy guide

Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.

What makes these hard

Why C is right

Why the others are wrong

Why C is right

Why the others are wrong

Why A is right

Why the others are wrong

Why B is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why C is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why B is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why B is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why A is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why A is right

Why the others are wrong

Why B is right

Why the others are wrong

Why C is right

Why the others are wrong

Why D is right

Why the others are wrong

Why A is right

Why the others are wrong

Why A is right

Why the others are wrong

Why C is right

Why the others are wrong

These 25 are a sample, not a study plan

More hard Geometry and Trigonometry questions